The integrated optical diodes have been a thriving research theme due to their potential on-chip applications in photonic circuits for all-optical computing and information processing. Analogous to electronic counterparts, the unidirectional light propagation is characterized by the high contrast between forward and backward transmissions. In this chapter, we demonstrate the proposed schemes and designs for reciprocal and nonreciprocal optical diodes based on two-dimensional (2D) photonic crystal (PhC). The reciprocal devices are built by linear and passive PhC, and the spatial asymmetric mode conversion is utilized to achieve the unidirectionality. The presented nonreciprocal optical diodes rely on the optical nonlinearity of cavity. New 2D PhC optical diodes with high contrast ratio, low insertion loss, large operational bandwidth, small device footprint, and ease of fabrication are highly desirable and still pursued.
- two-dimensional photonic crystal
- optical diode
- unidirectional transmission
- time-reversal symmetry
- mode conversion
- nonlinear effect
The optical diodes, as the counterpart of electron diodes in optics, have attracted huge interests due to its capability of asymmetric light propagation. The unidirectionality invokes potential applications in integrated optical circuits for all-optical computing and information processing [1, 2, 3, 4, 5]. Highly desirable key performances of an optical diode are high contrast ratio, low insertion loss, large operational bandwidth, small device footprint, and ease of fabrication. The two types of diodes, reciprocal and nonreciprocal optical diodes, have been proposed based on different mechanisms. The conventional nonreciprocal designs utilize magneto-optical effect [6, 7, 8] or nonlinearity [9, 10, 11, 12]. The reciprocity of the Lorentz theorem is broken in such designs. More importantly, the optical isolator which prohibits the propagation of any possible mode in one direction can be achieved . Meanwhile, the reciprocal optical diodes, in which the time-reversal symmetry holds, have drawn increasing attention in recent years. The underlying mechanism is the mode conversion in linear structures with spatial asymmetry . In such kind of the optical diode, the unidirectional transmission naturally exists for only certain modes. Due to the flexibility of designing coupling structures and tuning the photonic properties, the two-dimensional (2D) photonic crystal (PhC) has been shown as a promising building block to realize the optical diode. In this chapter, we will provide a summary of proposed schemes of both reciprocal and nonreciprocal optical diodes based on the 2D PhC slabs. This chapter is organized as follows: Section 2 will focus on the reciprocal optical diode based on the 2D PhC composed of dielectric rods or air-holed dielectric slab; Section 3 will introduce nonreciprocal optical diode based on nonlinear effects.
2. Reciprocal optical diode
Any reciprocal device based on passive and linear structure holds the time-reversal symmetry because the materials are described by scalar permittivity and permeability. The function of such reciprocal device has been proven and can be described by converting one set of orthogonal input modes to a matching set of orthogonal output modes . In waveguide structure, the conversion between low-order even and odd modes is usually utilized. For an ideal reciprocal optical diode, complete conversion of a specific mode in one direction and meanwhile complete reflection of the same mode in opposite direction are required. An example of optical diode is illustrated in Figure 1. Modes A and B are used to denote the even mode and odd mode, respectively. The even mode A incident from the left port is converted into odd mode B on the right, while the odd mode B from the left is completely reflected. Meanwhile, the transmission of even mode A is not allowed from right to left, while the odd mode B from the right is converted to even mode A. It can be seen that the diode effect for even mode is achieved in this device. The symmetric even-to-odd mode and odd-to-even mode conversions reflect the time-reversal symmetry. It should be noted that this linear and passive device cannot block all possible modes in one direction, which means it is not an optical isolator. In this section, we will focus on the proposed schemes and designs in the last decades, which are based on the 2D PhC composed of dielectric rods or air-holed dielectric slab.
2.1. Rod-type PhC optical diode
The rod-type 2D PhC represents the crystal consisting of periodic dielectric rods immersed in air. The rod material is commonly set as silicon in the below designs. The polarization of light is usually the chosen TM mode whose electric field is parallel to the rod. This is determined by the bandgap of such kind of PhC. The optical properties can be simulated by finite element method (FEM) or finite-difference time-domain (FDTD). Almost all simulations of rod-type designs discussed here are within two-dimensional. The length of rod is assumed infinite. Moreover, when estimating the performance of diode effect, two terms “unidirectionality” and “contrast ratio” may be used. The unit of unidirectionality is dB, and the expression is
Feng and Wang  proposed a device with the abilities of wavelength filtering and unidirectional light propagation based on the 2D square-lattice PhC with a rectangular defect. In the L-type device shown in Figure 2, the left W1-typed waveguide formed by removing one row of rods only sustained an even-mode guiding band spanning the frequencies from 0.3020 c/a to 0.4160 c/a. The under waveguide constructed by removing one row of rods and shifting the adjacent two rows of rods outward for the distances of 0.7a and 0.35a could hold the odd-mode guiding bands from 0.3308 c/a to 0.4160 c/a and even-mode guiding bands within the frequencies from 0.2746 c/a to 0.4160 c/a. The rectangular defect was located at the corner to connect two waveguides. The two side lengths of the rectangle were 1.367a and 0.5a, respectively, and the long side was placed along the x-direction of the PhC. There were four defective modes localized by this defect. The adopted mode was located at 0.3977 c/a, and the symmetry was odd in the horizontal direction and even along the vertical direction. The unidirectional propagation of the fundamental mode (even mode) for TM polarization was realized at 0.3977 c/a. The insertion loss was about −6 dB in forward direction, and about 25 dB unidirectionality was achieved. Moreover, the resonant frequency can be linearly tuned by adjusting the coupling region between the defect and the input waveguide. Due to the coupling between defect/cavity and waveguide, the operational bandwidth is narrow.
Ye et al.  realized the optical diode effect for even mode by a 2D PhC structure comprising two line-defect waveguides (W1, W2) and a simple defect cavity, as shown in Figure 3. Here, the numbers of rows for removed rods were 1 and 2 in W1 waveguide and W2 waveguide, respectively, and the cavity was made by removing two central rods. From left to right, the structure was divided into five parts: W2 waveguide, four-column length matching stage, W1 waveguide, cavity, and W2 waveguide. The mode conversion was impelled by the similarity between the odd mode in the W2 waveguide and the second cavity mode with even symmetry in the horizontal direction and odd symmetry in the vertical direction at the resonant frequency 0.3930 c/a. When the even mode was inputted from the left port at this frequency, it would couple to the second mode in the cavity and then coupled to odd mode in the W2 waveguide. The unidirectional propagation of even mode was achieved thanks to the W1-cavity-W2 design breaking spatial symmetry. The full width at half maximum of the forward transmission spectra of the proposed design with incidence of even mode was about 0.0002 c/a. The transmission efficiency reached a peak 92.5% at the resonant frequency in forward direction, while the value stayed below 0.1% within the frequency range from 0.3900 c/a to 0.3960 c/a for the backward transmission, and the unidirectionality reached approximately 35 dB at resonant frequency. The resonant frequency could be tuned by changing the radii of the nearest rods surrounding the defect and shape of the cavity according to the equation
where r denoted the radius of the nearest rods surrounding the defect and
Khavasi et al.  reported a broadband optical diode based on pure PhC waveguide with spatial asymmetry as illustrated in Figure 4. The components of the design mainly included a mode converter, a mode filter, a four-column length matching stage, and line-defect waveguides formed by removing two rows (W2) and one row (W1) of rods. Along the forward direction, the PhC structure could be divided into seven parts: W2, matching stage, W1, mode converter, W2, mode filter, and W2. The two different routes, U-shaped and straight paths, introduced the phase shifting in the mode converter with 5 × 5 unit cells. The geometrical lengths of the left, upper, and right arms of the U-shaped path were 3a, 2a, and 2a, respectively. And, two
Lu et al.  realized the optical diode effect based on two-dimensional square-lattice PhC, in which a directional coupler and 90° bend were utilized to realize the mode converter and mode filter functions. The scheme is depicted in Figure 5. The components in leftward direction were, respectively, W2 waveguide formed by removing two rows of rods, mode filter and mode converter, and W1 waveguide created by removing one row of rods, a four-column length adiabatic-matched region and W2. A perpendicular W1 waveguide and two 90° bends were combined together to achieve a mode converter. The conversion principle was as follows: when the light transmitted in the leftward direction, the odd mode was even symmetric along the horizontal direction in the mode filter, while the fundamental even mode in the perpendicular W1 waveguide was also symmetric about the parallel direction. Therefore, the odd mode could be thoroughly converted into the even mode after the 90° waveguide bend. On the other hand, the even mode would be converted into odd mode based on the same theory in the rightward direction. The mode filter was designed by placing a row of rods in the middle of W2 as several other works. For the device, the odd mode was completely transmitted in forward direction (left to right) and converted to the fundamental even mode but was blocked in the opposite direction. Approximate 95% forward transmission efficiency, 0.024 c/a operational bandwidth, and maximum 80 dB unidirectional extinction ratio were achieved in 2D simulations.
Kurt et al.  reported a photonic structure formed by using a chirped PhC waveguide to realize asymmetric light propagation as shown in Figure 6. A distance increment between each neighboring column of the dielectric rods, which caused that the waveguide mode moved toward the outside of the bandgap, was introduced to break the spatial symmetry. Here, the distance between each unit cell was still a constant along the y-direction (Δy = a), while the interval of rods linearly increased along the x-direction (0.5a < Δx < 2.5a), and the total length of the chirped PhC waveguide was approximately 16a. Thus, the structure became sparse along the x-direction, and the spatial distribution was uniform along the y-direction. When the light traveled in forward direction, the light mainly propagated at the center and leaked toward the two sides of the waveguide after propagating a section of distance; then, it traveled again along the x-direction and reached the end of the structure. The amount of light reaching the waveguide centerline was very small. However, the central lobe strongly appeared at the waveguide exit in the backward light transmission. The contrast ratio was around 0.75 at the operating frequency
Giden et al.  from the same research team as the above work theoretically and experimentally realized the asymmetric light transmission based on PhC with a gradient index distribution along the x-direction, in which the waveguide was formed by introducing point and line defects. The total length and width of the proposed structure were L = 16a and W = 20a, respectively. Continuously graded refractive index distribution could be approximated by relocating the PhC dielectric rods. In order to achieve a desired index profile within certain wavelength, they utilized an equation n(
Soltani et al.  proposed a 2D PhC made of elliptic silicon rods to realize the unidirectional propagation as depicted in Figure 7. The major and minor axes of the ellipse were 0.31a and 0.22a, respectively. The waveguide was created by removing one row of the dielectric rods along the x-direction. The distance between neighboring columns was modulated according to the equation,
Singh et al.  realized the optical diode effect based on a hexagonal and square-lattice PhC structures with silicon rods immersed in air. The structure is demonstrated in Figure 8. The waveguide was formed by removing one row of rods and chirping the neighboring rods, which was realized by changing the radii of the rods. Two kinds of devices were proposed according to the different lattice. For the regular hexagonal lattice PhC, the lattice constant a and radius of the rods r were 0.68 μm and 0.23 μm, respectively, and the radii of the chirped rods were
Liu et al.  devised an ultracompact mode converter based on a two-dimensional square-lattice PhC made of dielectric rods in air by a two-phase numerical optimization method and applied it to the optical diode. In numerical design and optimization, the design was evaluated by the error metric, , where Pω was the conversion efficiency at frequency ω. A region with rods located on the 20 possible lattice sites was considered to limit the design space in the optimization process. In the first phase, a combinatorial search is performed using simulation method to analyze every possible combination of rods (presence or absence) on the 20 lattice sites in the region. Any structures with desirable conversion behavior (
Ye et al.  demonstrated a compact broadband optical diode based on linear 2D rod-type PhC as shown in Figure 10. The proposed structure consisted of a simple waveguide made by removing two rows of rods and a 4 × 1 unit cells’ large functional region, the spatial dielectric distribution in which was numerically optimized by finite element method (FEM) combining the geometry projection method (GPM)  and the method of moving asymptotes (MMA) . Two dielectric rods inside the functional region were still kept during the optimization process. The GPM employed 6 × 20 control points whose heights can be fitted as a three-dimensional surface and the intersection line of level plane and three-dimensional surface could be used to define the rod shape. When level plane was set at zero, the dielectric constant at coordinate x could be formulated as , where S(x) was the surface function, d(x) was the minimum distance from the refined surface to the nearest intersection curve, and ξ was the control parameter of vanishing of intermediate dielectric at interface. The GPM controlled the rod shape, size, and position by adjusting the heights of the control points. The FEM and MMA were utilized to determine the heights of control points to find the solution. The optimization process proceeded in two phases to obtain large operating bandwidth. First, they searched a candidate at the center frequency 0.40 c/a; in the second phase, they optimized further by the new objective function, which was the weighted average of objective function values at five frequencies within the targeted frequency domain. The integrals of power flux for desired optical behavior were set as objective functions during the optimization process. The maximum forward transmission efficiency was 83%, and the above 60% transmission efficiency within bandwidth 0.01 c/a was achieved. The above 19 dB unidirectionality (maximum 30 dB) for even mode was achieved within bandwidth 0.01 c/a as well.
Wang et al.  demonstrated a graded photonic crystal (GPhC) design with 11a × 25a large scale, which was obtained from a conventional square-lattice PhC with
2.2. Air-holed PhC optical diode
Most of the designs based on rod-type PhC discussed above were confined in simulation because of the difficulties in fabricating. The only experimental validation was reported in Giden?s work , but operating in the microwave regime (10 GHz). Moreover, the absence of light confinement along the rod direction is another limitation. Compared with rod-type PhC, the fabrication technique of air-holed PhC slab is more mature and compatible with conventional CMOS processing. Air-holed PhC is formed by etching periodic air holes in dielectric slab. The optical diode based on air-holed PhC can be directly connected to standard silicon waveguide, showing its ability of integration with other silicon-based photonic devices. However, unfortunately, lack schemes for air-holed 2D PhC optical diode can be found. One scheme is based on the PhC heterostructure with directional bandgap [28, 29, 30, 31], and the other scheme is based on topology optimization . The TE mode whose magnetic field is perpendicular to the slab is considered in all designs discussed in this section and most simulations are performed in 3D FDTD.
Cicek et al.  realized the unidirectional light transmission in a 2D PhC heterostructure with air holes in an AlGaAs host. The proposed structure was constructed of two kinds of photonic crystals (PhC1, PhC2) with the same orientation and lattice constant but different radii of air holes, as illustrated in Figure 12. The interface was located along its body diagonals. The radius of air holes in PhC1 with 16 × 16 unit cells was set as 0.460a, while the value in PhC2 with 15 × 15 unit cells was 0.352a. Moreover, PhC2 was offset by
Wang et al. [29, 30] proposed an optical diode design with functional region 6 × 6 μm2 based on silicon PhC slab. The radii of air hole for PhC1 and PhC2 were 0.24
Ye et al.  realized the optical diode effect in 2D silicon PhC slab waveguide with 1.2a × 2.8a large functional region, in which there was only one deformed air hole. The structure is illustrated in Figure 14. The waveguide was constructed by removing one row of air holes along x-direction and moving the first and second nearest row with 0.15a and 0.1a in y-direction, respectively. The spatial dielectric distribution in the functional region was numerically optimized by finite element method (FEM) combining the geometry projection method (GPM) and the method of moving asymptotes (MMA). The GPM employed 13 × 29 control points to fit the three-dimensional surface. The optimization algorithm MMA was utilized to determine the heights of all control points in GPM. In order to find a broadband optical diode, three optimization steps were taken: first, they searched a bidirectional mode converter with mirror symmetry in functional region at the center frequency 0.36 c/a. Next, the symmetry was broken to find the solution for optical diode working at center frequency. The last step was to extend the operational frequency range from single frequency. Here, the integrals of power flux for desired optical behavior were set as objective functions. The above 89.9% forward transmission efficiency, 0.01 c/a operational bandwidth, and maximum 24 dB unidirectionality were achieved in 2D FEM simulations. In the air-bridged 3D FDTD model, the maximum unidirectionality was about 19.6 dB, while the value kept higher than 12.7 dB for slab thickness 0.54a. It should be noted that identical performance of 3D device and 2D optimized structure remains a challenge.
3. Nonreciprocal optical diode
The optical diode effect achieved in linear and passive PhC (discussed in Section 2) cannot break the reciprocity of the Lorentz theorem and consequently cannot play as an important optical device: the optical isolator. The conventional nonreciprocal designs utilize magneto-optical effect or nonlinearity. Unfortunately, common waveguide materials, especially Si, SiO2, and Si3N4, present only relatively weak or even no magneto-optical effect. Meanwhile, the prerequisite external magnetic field would limit the integration for on-chip applications. Therefore, we will focus on the nonlinearity in this section. The major effects may include the second-order optical nonlinearity, the Kerr effect, two-photon absorption, free-carrier effect, and thermo-optic effect . These effects have been pervasively employed in on-chip all-optical processing such as wavelength conversion, optical switch, modulation, and optical isolation [34, 35, 36].
Zhang et al.  proposed a silicon optical diode based on optical nonlinearity in cascaded PhC L3 cavities, which was formed by removing a line of three holes as demonstrated in Figure 15. The lattice constant was 420 nm, and the air-holed radius was 126 nm. The three air holes adjacent to the right cavity were shifted by 0.175
Bulgakov and Sadreev  proposed an all-optical diode in asymmetric L-shaped PhC waveguide with a single nonlinear Kerr microcavity at the corner. The structure is depicted in Figure 16. The linear L-shaped PhC waveguide was formed by removing one row of dielectric rods, and the microcavity was constructed by three linear rods and one nonlinear rod. The radius and dielectric constant of the GaAs rods were 0.18a and 11.56 in the PhC, where the lattice unit was set as 0.5 μm. The nonlinear defect rod had the radius 0.4a, and the dielectric constant was 6.5. The nonlinear refractive index was 2 × 10−12 cm2/W. In order to achieve the asymmetric design of the microcavity, the dielectric constant of the third additional rod inside the right leg of the waveguide was chosen as 5. The two dipole eigenfrequencies of the microcavity were 0.3658 × 2πc/a and 0.3650 × 2πc/a, respectively. When both dipole modes were excited simultaneously, the design could be opened for light transmission at some magnitudes of frequency and injected intensity because of the nonlinear coupling between them. The opening region mostly depended on the ratio between the nonlinearity constant and the coupling strengths of the dipole modes with the propagating waveguide modes. Therefore, the asymmetric transmission in the design was achieved due to the different threshold for light transmissions from left to right and backward. This came from different coupling strengths of dipole modes with the propagating mode of the left and right waveguide legs. In the design, the direction of optical diode transmission could be controlled by power or frequency of injected light.
Liu et al.  realized the optical diode effect based on asymmetrical coupling by a microcavity with nonlinear Kerr medium and a FP cavity at two sides of PhC waveguide. In the system, a Fano resonant microcavity with ultrahigh-quality factor was designed in rod-type PhC. An elliptical point defect was placed in the first layer below a line-defect PhC waveguide, and a FP cavity was located in the second layer above PhC waveguide, as shown in Figure 17. Moreover, a reflector layer was set below the left end of FP cavity to form asymmetric structure. The radius of dielectric cylinder was 0.3a; the major and minor axes of elliptical point defect were set as 0.534a and 0.3a, respectively. Because of the asymmetric design, backward (right to left) propagation required stronger incidence light to excite Kerr effect of microcavity than forward (left to right) transmission, so the unidirectional transmission could be realized. The high transmission in forward direction was approximately about 80%, and maximum contrast ratio was about 0.8.
In this chapter, the proposed schemes for optical diodes based on 2D PhC have been briefly summarized. For reciprocal optical diode, we introduce 11 designs based on rod-type PhC and 4 designs based on air-holed PhC slab. Since the linear and passive structures hold the time-reversal symmetry, the unidirectional transmission of certain optical mode is achieved by spatial asymmetric mode conversion. For nonreciprocal optical diode, we mainly focus on the PhC cavity with optical nonlinear effects. The three designs can be used as optical isolator since it can block all possible modes in one direction. Considering the performances, new schemes for 2D PhC optical diodes with high contrast ratio, low insertion loss, large operational bandwidth, small device footprint, and ease of fabrication are still highly desirable.